The laws of orbital motion are mathematical, and one cannot explore them without some mathematics. The math used here is rather elementary; if you need a refresher click here. Otherwise, you can just skip the equations and follow the narrative.

The Mathematical Description of a Curve

As already noted, the cartesian system labels any point in a plane (e. g. on a flat sheet of paper) by a pair of numbers (x,y), its distance from two perpendicular axes. These numbers are known as the "coordinates" of the point.

A line in the plane--straight or curved--contains many points, each with its own (x,y) coordinates. Often there exists a formula ("equation") which connects x and y: for instance, straight lines have a relationship

y = ax + b

where any pair of numbers (a,b), positive, negative or zero, gives some straight line. The plot of a line given by one of such relationship (or indeed by any relationship--even pure observation, e.g. temperature against time--is known as a graph. More complicated relationships give graphs that are curves: for instance

y = ax2

gives a parabola, with a any number. Usually (though not always) y is isolated, so that the formula has the form

y = f(x)

where f(x) stands for "any expression involving x" or in mathematical terms, a "function of x." The curves drawn here are the straight line y = -(2/3)x + 2 and the parabola y = x2. A list of some of their points follows.

Straight line:

x

-1

0

1

2

3

4

y

8/3

2

4/3

2/3

0

-2/3

Parabola:

x

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

y

4

2.25

1

0.25

0

0.25

1

2.25

4

The Equation of a Circle

In the vast majority of formula-generated graphs, the formula is given in the form

y = f(x)

Such a form makes it very easy to find points of the graph. All you have to do is choose x, calculate f(x) (= some given expression involving x) and out comes the corresponding value of y.

However, any equation involving x and y can be used as the property shared by all points of the graph. The main difference is that with more complicated equations, after x is chosen, finding the corresponding y requires extra work, (and sometimes it is easier to choose y and find x).

Perhaps the best-known graph of this kind is a circle of radius R, whose equation is

x2 + y2 = R2

Draw a circle of radius R centered at the origin O of a system of (x,y) axes . Given any point P on the circle with specified values of (x,y), draw a perpendicular line from P to point A on the x-axis. Then

x = OA y = AP R = OP

Here x and/or y may be negative, if they are to the left of the y-axis or below the x-axis, but regardless of sign, x2 and y2 are both always positive. Since the triangle OAP has a 90° angle in it, by the theorem of Pythagoras, for any choice of P, the relation below always holds:

OA2 + AP2 = OP2

Since this can also be written

x2 + y2 =
R2

The equation of the circle is satisfied by any point located on it. For instance, if the graph is defined by the equation:

x2 + y2 = 25

this equation is satisfied by all the points listed below:

x

5

4

3

0

-3

-4

-5

-4

-3

0

3

4

( 5 )

y

0

3

4

5

4

3

0

-3

-4

-5

-4

-3

( 0 )

The Equation of an Ellipse

The equation of the circle still expresses the same relation if both its sides are divided by
R2:

(x2/R2) +
(y2/R2) = 1

The equation of an ellipse is a small modification of this:

(x2/a2) +
(y2/b2) = 1

where (a,b) are two given numbers, for example (8,4). What would such a graph look like? Near the x axis, y is very small and the equation comes close to

(x2/a2) = 1

From which

x2 = a2 and hence x = a or x = –a (sometimes combined into x = ±a)

The graph in that neighborhood therefore resembles the section of a circle of radius a, whose equation

(x2/a2) +
(y2/a2) = 1

also comes close to x2 = a2 in this region. In exactly the same way you can show that near the y-axis, where x is small, the graph cuts the axis at y=±b and its shape there resembles that of a circle of radius b.

An example

Let us draw the ellipse

(x2/64) +
(y2/16) = 1

We already know that it cuts the axes at x=±8 and at y=±4. Let us now add a few points:

(1) Choose y = 2 .
Then from the equation

(x2/64) +
(4/16) = 1

Substract 1/4 from both sides

(x2/64) =3/4

Take roots (marked here by √) and retain only 3-4 figures:

x/8 = √3 / √4 = 1.732/2 = 0.866

from which x = 6.93 within resonable accuracy.

(2) Choose y = 3 .

(x2/64) +
(9/16) = 1

Substract 9/16 from both sides

(x2/64) =7/16

Take square roots (to an accuracy of 3-4 figures):

x/8 = √7 / √16 = 2.6457/4 = 0.6614

from which, approximately, x = 5.29

Again, either sign can be attached to x and y. We get 12 points, enough for a crude graph: